MA101.16 Functions of two variables or more

For a function of two (real) variables each element of its domain is an ordered pair of real numbers (x,y) only, and each element of its range is a real number, z, say.

x and y are called independent variables, and z the dependent variable.

To specify such functions we must have both a rule, and a domain.

Example

i) \begin{aligned} z=\sqrt{1-x^2-y^2} \qquad&\mbox{domain: }&&\{(x,y) | x^2 +y^2 \leq 1\} \\&\mbox{range: }&&[0,1]\end{aligned}

ii) \begin{aligned} f(x,y)=\sqrt{2-x}+\sqrt{9-y^2} \qquad&\mbox{domain: }&&\{(x,y) | x \leq 2, -3\leq y\leq3\} \\&\mbox{range: }&&[0,\infty)\end{aligned}

If no known domain is specified we again take it to be the largest subset of \mathbb{R}^2 for which the rule makes sense.

A graph of z=f(x,y) may be drawn in the usual way. (ie. (x,y) in the horizontal plane called the xy plane, and z moving along a third, vertical axis.

For obvious reasons informative graphs are often quite difficult to draw. It’s often useful to draw profiles of f, that is the surface curves where their surface meets planes parallel to the coordinate axis.

Example

Sketch x=e^{-xy}

Consider the intersection with x=constant, y=constant and z=constant.

\begin{aligned} x=0 \qquad&z=1 \qquad\qquad&&y=0 &&&&z=1\\x=1 \qquad&z=e^{-y} \qquad\qquad&&y=1 &&&&z=e^{-x}\\x=2 \qquad&z=e^{-2y} \qquad\qquad&&y=2 &&&&z=e^{-2x} \end{aligned}

z=\frac{1}{2} \Rightarrow \frac{1}{2}=e^{-xy} \Rightarrow e^{xy}=2 \Rightarrow xy=\ln{2} \\  z=e^{-1} \Rightarrow e^1 = e^{xy} \Rightarrow xy=1

Another useful diagram of f(x,y) is that provided by drawing the contour lines (or level curves). That is we sketch in the x,y plane the graph of y=f(x,y) for various values of z.

Look at z=e^{-xy} again.

-xy = \ln{z} so xy =-\ln{z}

When,

\begin{aligned}&z=1 \qquad&&xy=0 \\&z=e&&xy=-1 \\&z=e^4&& xy=-4 \\&z=e^{-1}&&xy=1 \\&z=e^{-4}&&xy=4 \end{aligned}

Example

Sketch z=x^2+y^2

For a function w=f(x,y,z) we can not sketch profile, and the best we can do is sketch level surfaces.

Example

w=x+y+z

For w=2 \Rightarrow 2-x-y=z etc…